Date: Wed, 24 Mar 1993 10:44 EST
From: winkel at nextwork.rose-hulman.edu
Concerning Michael Ibrahim's question of making Mathematica output
look like the textbook solutions for trig integrals, I say, "Let us
not worry about that!"
[...]
Shouldn't one consider the specifics of Michael's question before saying
that? If Mathematica may be unnecessarily giving especially foolish
answers from a mathematical point of view, shouldn't that be addressed?
Here, repeated, are Michael's examples:
============================================
In[1]:= Integrate[(1+Sin[x])/Cos[x]^2,x]
x
2 Sin[-]
2
Out[1]= ---------------
x x
Cos[-] - Sin[-]
2 2
In[2]:= Integrate[(1+Sin[x])/Cos[x]^2,{x,0,Pi/4}]
5/4
(1 + I) (1 + (-1) )
Out[2]= ---------------------
1/4
-I + (-1)
============================================
I find it troublesome that Mathematica is introducing half angles for
a problem that can easily be done without them. Returning answers
using the same types of functions as in the input makes the answers
more easily comprehensible, and easier to confirm by differentiation
and simplification. E.g. Macsyma returns tan(x) + 1/cos(x) for the
problem in In[1].
But I find more surprising that you can easily accept the answer in
Out[2] as a substitute for the easily obtained Sqrt[2] ! The answer
returned looks incomprehensible to me, unnecessarily involves complex
quantities, and isn't even simplified ( (-1)^(5/4) is just -(-1)^(1/4) ,
not to mention -Sqrt[I] .)
When the zero was introduced and the Roman
numeral system was challenged, I doubt if the philosophy of those
wanting to move ahead with a more capable system of counting always
tried to get the new number writing system to look like Roman system.
We need to move on.
What's wrong with examining whether the computer algebra systems we
use are doing good, correct, comprehensible symbolic mathematics?
From: Jeffrey P. Golden <jpg at macsyma.com>
Organization: Macsyma Inc.